Leonard, I. E.; Taylor, K. F. Essential supremum norm differentiability. (English) Zbl 0593.46016 Int. J. Math. Math. Sci. 8, 433-439 (1985). The main results of this paper are the following Theorem 3.1. Let \((\Omega,\Sigma,\mu)\) be a finite measure space, X a Banach space, and \(f\in L_{\infty}(\mu,X)\) with \(f\neq 0\). Then f is a smooth point of \(L_{\infty}(\mu,X)\) iff there exists an atom \(A_ 0\) for \(\mu\) such that (1) \(\| f\| >ess \sup \{\| f(\omega)\|:\omega \in \Omega \setminus A_ 0\}\), and (2) \(x_ 0\) is a smooth point of X, where \(x_ 0\) is the essential value of f on \(A_ 0.\) Corollary 3.2. The norm on \(L_{\infty}(\mu,X)\) is Fréchet differentiable at f iff there exists an atom \(A_ 0\) for \(\mu\) such that (1) \(\| f\| >ess \sup \{\| f(\omega):\omega \in \Omega \setminus A_ 0\}\), and (2) the norm on X is Fréchet differentiable at \(x_ 0.\) An application of these results is given to the space of all bounded linear operators from \(L_ 1(\mu,{\mathbb{R}})\) into X. Reviewer: M.Kadets Cited in 1 Document MSC: 46B20 Geometry and structure of normed linear spaces 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E40 Spaces of vector- and operator-valued functions 47B38 Linear operators on function spaces (general) Keywords:essential supremum norm differentiability; smooth point; Fréchet differentiable; space of all bounded linear operators PDFBibTeX XMLCite \textit{I. E. Leonard} and \textit{K. F. Taylor}, Int. J. Math. Math. Sci. 8, 433--439 (1985; Zbl 0593.46016) Full Text: DOI EuDML